Belief

The Bayesian Theorem is one of the most powerful black magic tools of mathematical occultism. I want to formulate in plain text the purpose and properties of Bayesian Theorem.

Given an evidence, to what degree can we assume a hypothesis is true?

Lets assume

Evidence e: a bettor places a for him unusual high amount on a horse

Hypothesis h: a bettor has inside information

Problem p(e|h): When we know e, to what degree can we assume h?

Hypothesis h is a little hard to prove. We would need a court to judge that there was a inside information or even manipulation of the game. Lets relax h to a more broader term and reformulate the problem. When a bettor places a unusual high amount what is the chance that he wins? We relax h to:

Hypothesis h1: a bettor wins
Where h is part of h1, i.e. a insider needs to win to be an insider.

We could look at what we know about our bettors:

1. Certainly we know when a bettor places an unusual high amount, lets say more that 5 times the average in a similar bet. We know how many bettors do that, p(e).
   
2. We know how many bettors are winning in our horse races, p(h1).
3. We know how many past times winners betted high, p(e|h1) means literally: given that a person wins what is the likelihood he betted high? We know that from the past.

Now Bayes says we can predict from this data if a person will win if he is staking high.

p(h1|e)= p(e|h1) p(h1)/p(e)

Lets say p(e|h1)>0.5, then we know winners are usually staking high. This gives us evidence that they are confident to win. Why could that be? Reflect on this :-)

p(h1) in a game like horse race should not exceed 0,5 since all the money is in the pot, i.e. there are no more winners than losers and winners are just a few. p(e) will be also under 50%, just because otherwise the amount would be "usual". Whats important here: the smaller p(e) in comparison to p(e|h1) p(h1), the greater p(h1|e).

The chance a highstaker is winning rises when fewer people are betting high amounts, while the ratio of winners that betted high stays the same. That is certainly true.

What is when p(h1|e)>0.5? Then its likely that high stakes lead to big wins.

In the case of p(h1|e)<0.5 I would certainly bet with the bettor without spending a doubt on his reputation.